# Tagged Questions

Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

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### Is there a known mathematical equation to find the nth prime?

I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?
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### Can this function be a new test for primality?

The following function returns always 0 only if a number is not prime. $$H(x)=\prod_{i=2}^{x-1}\left\{\left[\sum_{k=1}^{x/i}(-i)\right]+x\right\}$$ what do you think? Bye!
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### the least prime leaving prime remainder when divided by 2,3,5,…

What is the least prime giving a prime remainder when divided by 3? It is 5. What is the least prime giving prime remainders for both 3 and 5? It is 17. For division by 3,5,7 it is also 17. For ...
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### It is possible to find the exact value of prime number without approximation ( PNT)? For instance I want to know how many primes are below 8111.

It is possible to find the exact value of prime number without approximation ( PNT)? For instance I want to know how many primes are below 8111 or any value of N.
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### Testing prime numbers with modified Fermat's Little Theorem

Is there a number $n$ such that: $6n-1$ is prime There exists a positive integer $r<3n-1$ such that $4^{r}\equiv1\pmod{6n-1}$
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### Using the prime number theorem to find a continuous function mapping primes?

The prime number theorem gives an increasingly (proportionally) accurate approximation for the number of primes below $x$. Can we use this to find an equivalently accurate approximation which maps the ...
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### $\sup\left(\frac{\log(\mbox{ lcm }(1,2,\ldots,k))}{k}\right)$ for $k\in \Bbb{Z}, k>1$

In a previous question Asymptotic growth of l.c.m. of all integers below $k$, it was noted that using the Prime Number Theorem you can prove that $$\log(\mbox{ lcm }(1,2,\ldots,k)) =k+\mbox{ o}(k)$$ ...
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### Are there infinitely many pairs of primes where each divides one more than the square of the other?

I have the following question on number theory that is eating my head. Are there infinitely many primes $p,q$ such that $p | (q^2 + 1)$ and $q | (p^2 + 1)$? I can see $13,5$ and $2,5$ has the ...
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### Any prime number $(p)$, in sequence $(p^n, p^n+1…)$. Each term in sequence is divisible only for previous terms? [on hold]

Any prime number $(p)$, in sequence $(p^n, p^n+1...)$. Each term in sequence is divisible only for previous terms? This é relevant or simple derivation?
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### Small primes congruent to $a$ mod $p$.

Let $p$ be a prime and $a$ be an integer such that $0 \lt a \lt p$. Is there a prime number, $q$, congruent to $a$ mod $p$ such that $q\lt p^2$? I have checked that this is true for the first $3000$...
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### Euclid Mullin Sequence

Consider the Sequence as follows. Let $a_1 = 2$, $a_n$ be the largest prime divisor of $P_n = 1 + {\prod_{i = 1}^{n - 1} a_{i}}$ Then we obtain a sequence of prime numbers How do you show that 5 ...
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### Examples of Weil's explicit formula

In Bombieri, PROBLEMS OF THE MILLENNIUM: THE RIEMANN HYPOTHESIS, Clay Mathematics Institute (2000), from page 8, V. Further evidence: the explicit formula the author tell us that there is a ...
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### Which primes $p$ divide $q^q-1$ for a prime divisor $q$ of $p-1$

I am looking for (a formula) for all the primes $p$ less than or equal to $X$ with the following criteria: There is at least one prime $q$ dividing $p-1$ such that $p$ divides $q^q-1$. $7$, for ...
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### For each prime $p>3$ there are non twin primes $q,r$ with $p^3=2q+r$

Define $\mathbb P'=\{n\in\mathbb P|n-2,n+2\notin \mathbb P\}$. Conjecture: Given a prime $p>3$, then $\exists q,r\in\mathbb P':p^3=2q+r.$ Tested for the first 10000 primes. The solutions ...
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### Maximum length of a string that has no substring divisible by a prime number $p$ is $p-1$?

What is the maximum length of a string of nonzero digits that has no substring that is divisible by a given prime number? I want to find a string of length n which has no substring divisible by the ...
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### multiples (of primes) coverage formula

I apologize in advance if my explanation is not clear. Please let me know if clarification is required and I will do my best to fix it! I am attempting to find an explicit formula (in terms of ...
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### Conjecture: Every prime number is the difference between a prime number and a power of $2$

Conjecture: $\forall p\in\mathbb P\exists q\in\mathbb P\exists n\in \mathbb N: q-p=2^n$ Verified for the 100 first primes.